Determining a characteristic of a seismic sensing module using a processor in the seismic sensing module

ABSTRACT

A seismic sensing module comprises a seismic sensing element and a processor configured to generate a test signal applied to the seismic sensing element, receive a response from the seismic sensing element, and determine a characteristic of the seismic sensing module according to the response.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the earlier effective filing date of co-pendingProvisional U.S. patent application Ser. No. 60/981,141, filed on 19Oct., 2007, having the same title, by the same inventors.

TECHNICAL FIELD

The invention relates to determining a characteristic of a seismicsensing module using a processor in the seismic sensing module.

BACKGROUND

Seismic surveying is used for identifying subterranean elements, such ashydrocarbon reservoirs, fresh water aquifers, gas injection reservoirs,and so forth. In performing seismic surveying, seismic sources andseismic sensors can be placed at various locations on an earth surface(e.g., a land surface or a sea floor), or even in a wellbore, with theseismic sources activated to generate seismic waves. Examples of seismicsources include explosives, air guns, acoustic vibrators, or othersources that generate seismic waves.

Some of the seismic waves generated by a seismic source travel into asubterranean structure, with a portion of the seismic waves reflectedback to the surface (earth surface, sea floor, or wellbore surface) forreceipt by seismic sensors (e.g., geophones, hydrophones, etc.). Theseseismic sensors produce signals that represent detected seismic waves.Signals from the seismic sensors are processed to yield informationabout the content and characteristics of the subterranean structure.

A seismic sensor is typically tested. Conventionally, use of a highlyaccurate signal generator, which is separate from a seismic sensor, isusually required. The presence of the high accuracy signal generatoradds complexity and cost to a seismic survey system.

SUMMARY

In general, according to an embodiment, a method of testing a seismicsensing module includes generating, using a local processor associatedwith a seismic sensing element in the seismic sensing module, a testsignal that is applied to the seismic sensing element, and receiving, bythe local processor, a response of the seismic sensing element to thetest signal. The local processor determines at least one characteristicof the seismic sensing module, where the at least one characteristic isselected from a polarity of the seismic sensing element in the seismicsensing module, a non-linear property of the seismic sensing module, anda temperature-dependent transfer function of the seismic sensing module.

In general, according to another embodiment, a self-contained seismicsensing module includes a single housing that contains a seismic sensingelement, and a processor. The processor is configured to generate a testsignal applied to the seismic sensing element, receive a response fromthe seismic sensing element, and determine a characteristic of theseismic sensing module according to the response.

Other or alternative features will become apparent from the followingdescription, from the drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example marine survey arrangement.

FIG. 2 is a block diagram of a self-contained seismic sensing moduleaccording to an embodiment.

FIG. 3 is a graph of curves representing step responses over time tovarious input step voltages.

FIG. 4 is a graph of curves representing measured step response data ata certain time and an ideal step response.

FIG. 5 is a graph of a curve representing a step response with the idealstep response subtracted.

FIGS. 6A-6B illustrate different polarity connections to a seismicsensing element.

FIGS. 7A-7B and 8A-8B are graphs of curves representing positive stepresponses and negative step responses for different polarity connectionsof the seismic sensing element.

FIGS. 9A-9B are graphs of curves representing a reference accelerationfor use in developing a temperature-dependent transfer function.

FIGS. 10A-10B are graphs of curves representing the change in transferfunction with temperature at one frequency.

DETAILED DESCRIPTION

In the following description, numerous details are set forth to providean understanding of the present invention. However, it will beunderstood by those skilled in the art that the present invention may bepracticed without these details and that numerous variations ormodifications from the described embodiments are possible.

FIG. 1 illustrates an example arrangement to perform marine seismicsurveying. In different implementations, however, other embodiments caninvolve seabed seismic surveying, land-based seismic surveying orwellbore seismic surveying. FIG. 1 illustrates a sea vessel 100 that hasa reel or spool 104 for deploying a streamer 102 (or multiple streamers102), where the streamer 102 is a cable-like carrier structure thatcarries a number of electronic devices 103 for performing a subterraneansurvey of a subterranean structure 114 below a sea floor 112. In thefollowing, the term “streamer” is intended to cover either a streamerthat is towed by a sea vessel or a sea bed cable laid on the sea floor112.

The electronic devices 103 can include sensing modules, steering ornavigation devices, air gun controllers (or other signal sourcecontrollers), positioning devices, and so forth. Also depicted in FIG. 1are a number of signal sources 105 that produce signals propagated intothe body of water 108 and into the subterranean structure 114. Althoughthe sources 105 are depicted as being separate from the streamer 102,the sources 105 can also be part of the streamer 102 in a differentimplementation.

The signals from the sources 105 are reflected from layers in thesubterranean structure 114, including a resistive body 116 that can beany one of a hydrocarbon-containing reservoir, a fresh water aquifer, agas injection zone, and so forth. Signals reflected from the resistivebody 116 are propagated upwardly toward the sensing modules of thestreamer 102 for detection by the sensing modules. Measurement data iscollected by the sensing modules, which can store the measurement dataand/or transmit the measurement data back to a control system (orcontroller) 106 on the sea vessel 100.

The sensing modules of the streamer 102 can be seismic sensing modules,such as hydrophones and/or geophones. The signal sources 105 can beseismic sources, such as air guns, vibrators, or explosives. Seismicdata recorded by the seismic sensing modules on the streamer areprovided back to a control system (controller) 106 on the sea vessel.The control system 106 can process the collected seismic data to developan image of the subterranean structure 114.

In accordance with some embodiments, the seismic sensing modules caneach be locally associated with test circuitry to perform a test on therespective seismic sensing module so that a characteristic of thesensing module can be determined, where the characteristic can includenon-linear properties of the sensing module, a polarity of the sensingmodule, and/or a temperature-dependent transfer function of the sensingmodule. In some embodiments, the test circuitry can be integrated withthe sensing module such that a self-contained sensing module isprovided. More generally, test circuitry is “locally associated” withthe seismic sensing module if the test circuitry is located proximatethe seismic sensing module (rather than being located at a relativelyfar location such as on the sea vessel 100).

FIG. 2 shows an example embodiment of a self-contained sensing module200, which has an external housing 202 that contains various components.The components contained in the housing 202 of the sensing module 200include a seismic sensing element 204, such as a moving coil geophone,accelerator geophone, or other type of seismic sensing element. Inresponse to a step input (input test signal), represented as Ustep,which is generated during a testing procedure of the sensing module 200,the geophone 204 produces an output that is amplified by an amplifier206. The amplifier 206 produces an output signal representing the stepresponse, y(t), that is responsive to the step input Ustep.

The output signal representing y(t) is provided to the input of ananalog-to-digital (A/D) converter 208, which converts the analog outputsignal representing y(t) to digital data. The digital step response isthen processed by a processor 210. The processor 210 can be implementedwith a digital signal processor (DSP), a general purpose microprocessor,or any other type of processing element.

The step input Ustep is generated based on an applied input voltage U,provided by the processor 210 directly or indirectly (through othercircuitry). The processor 210 also controls geophone test switches 212Aand 212B, where the geophone test switch 212A connects the input voltageU to one side of the seismic sensing element 204, and the other geophonetest switch 212B connects the other side of the seismic sensing element204 to a reference voltage, such as ground. The processor 210 alsocontrols another set of switches 214A, 214B, which connect the output ofthe geophone 204 to the input of the amplifier 206. The switches 214A,214B are referred to as amplifier switches.

Note that during normal operation, which is operation of the sensingmodule 200 in the field for performing a seismic survey, the geophonetest switches 212A, 212B remain open, whereas the amplifier switches214A, 214B remain closed. In this “normal” configuration, the seismicsensing element 204 is able to detect a seismic input, such as in theform of an acoustic wave reflected from the subterranean structure, toproduce an output representing acceleration. The acceleration includesthe second derivative with respect to time of ground displacement, orfirst derivative of the velocity.

However, during a test operation for testing the sensing module 200, theamplifier switches 214A, 214B are initially open to isolate the outputof the seismic sensing element 204 from the input of the amplifier 206.Moreover, the geophone test switches 214A, 214B are also initially opensuch that no input is applied to the seismic sensing element 204. Toapply the input step, Ustep, to the seismic sensing element 204, thegeophone test switches 212A, 212B are closed. Note that the timeconstant of the switches 212A, 212B is much smaller than a time constantof the seismic sensing element 204 (in other words, the response time ofthe test switches 212A, 212B is much faster than the response time ofthe seismic sensing element 204). Simultaneously, or almostsimultaneously, with the closing of the geophone test switches 212A,212B (to apply the input step signal Ustep), the amplifier switches214A, 214B are also closed. Note that the response time of the amplifierswitches 214A, 214B is also much faster than the response time of theseismic sensing element 204. Thus, by the time that the seismic sensingelement 204 has responded to application of the input step signal Ustep,the amplifier switches 214A, 214B are already closed to allow the outputof the seismic sensing element 204 to be provided to the input of theamplifier 206.

To provide fast response times, the switches 212A, 212B and 214A, 214Bcan be implemented with solid state switches, such as transistors.

The input test voltage U can remain fixed during the entire duration ofa test procedure, or alternatively, the input test voltage U can bevaried by the processor 210. U can be varied to give constant currentthrough the geophone, e.g., when the geophone resistance changes. Thestep voltage, Ustep, is then measured separately. In one example, theprocessor 210 may be coupled to a temperature sensor 216 in the seismicsensing module 200 (or alternatively, to a temperature sensor locatedexternally to the sensing module 200) to receive temperature dataregarding an environment of the sensing module 200. The processor 210can vary the input test voltage U based on the temperature measurement,since the processor 210 may have to take into account variations in theresponse of the seismic sensing element 204 due to temperaturevariation.

From the step response produced by the seismic sensing element 204 as aresult of the input test signal, the processor 210 determines acharacteristic of the sensing module 200. The determined characteristiccan include a polarity of the seismic sensing element 204, atemperature-dependent transfer function of the seismic sensing module200, and/or a characterization of non-linear properties in a signalacquisition chain of the seismic sensing module 200 including theseismic sensing element 204, the amplifier 206, the A/D converter 208,and so forth.

Information regarding the determined characteristics can be stored in astorage 218 (e.g., memory, persistent storage, etc.) that is in thesensing module 200. Note that the storage 218 can be part of theprocessor 210. Also, in some cases, the processor 210 is able tocommunicate information regarding the determined characteristics througha network interface 220 (located inside the seismic sensing module 200)to an external network 222. Note that the external network 222 can bepart of the streamer 102 depicted in FIG. 1. The external network 222can be implemented with an electrical cable, a fiber optic cable, awireless communication medium, and so forth.

The network interface 220 in the seismic sensing module 200 includesvarious protocol layers to allow for communication over the externalnetwork 222, including a physical layer, data link layer, and higherlayers. In one example implementation, the network interface 220 caninclude Transmission Control Protocol (TCP)/Internet Protocol (IP)layers to allow for communication of control signals and data in TCP/IPpackets over the external network 222. In other implementations, theexternal network 222 can be a simpler network, such as a network thatincludes a control line and a data line. Also, the external network 222can be considered to include a power line to provide power to thesensing module 200.

The self-contained seismic sensing module 200 of FIG. 2 integrates aseismic sensing element (204) with test circuitry that is implementedwith the test switches 212A, 212B, amplifier switches 214A, 214B, andthe local processor 210, in the illustrated example. Integrating thetest circuitry with the seismic sensing element within the housing 202of the seismic sensing module 200 enables for more efficient andconvenient testing of the seismic sensing module 200. Note that thevarious components that are contained within the seismic sensing module200 can be provided on a circuit board. For example, the switches 212A,214B, 214A, 214B, amplifier 206, A/D converter 208, processor 210,storage 218, and network interface 220 can be implemented with one ormore integrated circuit chips that are mounted on the circuit board,where the circuit board is contained within the housing 202 of theseismic sensing module 200.

Non-Linearity Testing

As noted above, one of the characteristics of the seismic sensing module200 that can be determined includes non-linear property(ies) of a signalacquisition chain inside the sensing module 200. A seismic sensingmodule is considered to be non-linear if a response of the sensingmodule varies non-linearly with different amplitudes of an input testvoltage. For example, FIG. 3 shows a number of (non-linear) stepresponses, y(t), for different input step voltages, Ustep. A stepvoltage steps from an initial voltage level (e.g., 0 volts) to a second,different voltage level, which is indicated in the legend 300 in FIG. 3,where the example step voltages include +0.075, +0.10, +0.125, and soforth. The curves depicted in FIG. 3 represent the output signal voltageproduced by the seismic sensing element 204 as a function of time. Thus,for example, a first curve 302 represents the step response y(t)responsive to a step voltage of +0.075 volts (the input step voltagesteps from 0 to +0.075 volts). Similarly, another curve 304 representsthe step response, y(t), responsive to a step voltage of +0.55 volts(the input step voltage is stepped from 0 to +0.55 volts). Although notdepicted in FIG. 3, the input step voltages can also be negative stepvoltages, which means that the input test voltage steps from 0 to anegative voltage.

The step response y(t) is a function both of time and the applied inputstep voltage, U, i.e., y=y(t,U). The step response y(t,U) varies closeto linearly with U. Therefore, an assumption can be made that y(t,U) canbe approximately described by a Taylor series, i.e., a polynomial:

y(t,U)≈h ₀(t)+h ₁(t)U+h ₂(t)U ²+ . . . .  (Eq. 1)

In FIG. 4, the step response y(t), is shown as a function of an inputstep voltage U for the time t=0.1579 s in the example. Each circle inFIG. 4 represents an experimental data point (voltage value at theparticular U at time t=0.1579 s). The experimental data points in FIG. 4are fitted (third order polynomial fit) to a curve 402 (dashed curve).FIG. 4 also depicts a straight line 404 that represents an ideal linearstep response. Note that there is a slight difference between curves 402and 404 due to non-linearities of the seismic sensing module 200.

FIG. 5 shows the output signal voltage of the sensing module 200 withthe straight line (404 in FIG. 4) subtracted from the data (in otherwords, FIG. 5 shows just the non-linear part of the response). Thecircles in FIG. 5 indicate measured data values minus the straight line404 of FIG. 4, while the curve 502 in FIG. 5 is a third order polynomialfit of the data values depicted in FIG. 5. The curve 502 represents thecurve 402 of FIG. 4 minus the straight line 404.

The non-linearity of the seismic sensing module 200 is the differencebetween the measured data points (as fitted onto dashed curve 402) andthe solid straight line 404 (one example of a linear response of theseismic sensing module). In one implementation, the non-linearity of thesensing module 200 can be expressed using some predefined measure ofnon-linearity, such as a measure INL that is expressed as:

$\begin{matrix}{{{INL} = {\frac{1}{M}{\sum\limits_{m = 1}^{M}{\frac{1}{N}{\sum\limits_{n = 1}^{N}{{{y^{\exp}\left( {t_{m},U_{n}} \right)} - {y^{lin}\left( {t_{m},U_{n}} \right)}}}}}}}},} & \left( {{Eq}.\mspace{14mu} 2} \right)\end{matrix}$

Note that FIG. 4 represents y(t,U) as a function of different U valuesat a specific fixed time t. The example of FIG. 4 is for time t=0.1579s. FIG. 4 can be repeated for other time points (which correspond to thetime points along the time axis of FIG. 3). If FIG. 4 represents theresponse y(t₁,U) at time point t₁, then the other step responses atdifferent time points can be expressed as y(t₂, U), y(t₃, U), . . . ,y(t_(M), U), where M is an integer representing the number of timepoints that are considered.

In determining the non-linearity measure INL, the absolute value of thedifference between the fitted curve 402 (e.g., third-order polynomialfit), represented as y^(exp)(t_(m),U_(n)), and the straight line 404,represented as y^(lin)(t_(m),U_(n)), is determined and summed over U_(n)values for n from 1 to N. The sum of the difference values is divided byN to determine an average over the multiple U_(n) values.

Note that Eq. 2 also specifies summation over different time points,t_(m) for m=1 to M, with the summation of the differences over differenttime points divided by M to average over the number of time pointstaken. Effectively INL is calculated based on summing differencesbetween y^(exp) and y^(lin) over multiple (N) input step voltages U andmultiple (M) time points t, and dividing the sum by M×N to obtain anaverage of the differences.

The non-linearity measure INL can be stored in the storage 218 (FIG. 2)of the sensing module 200, or the non-linearity measure INL can becommunicated over the external network 222 to a central controller, suchas control system 106 in FIG. 1. If stored in the sensing module 200,the processor 210 is able to perform self-calibration of the measurementtaken with the sensing module 202 to compensate for non-linearproperties of the sensing module 200.

The following describes a more specific implementation for calculatingan INL measure to represent non-linearity of a seismic sensing module.The step voltages are denoted U₁, U₂, . . . , U_(n), . . . , U_(N). Foreach step voltage, a step response is measured: y(t_(m),U_(n)), wheren=1, 2, . . . , N and t_(m) is the time and m=1, . . . , M is the numberof samples.

For each m, the experimental data is fitted to a third-order polynomial:

y _(fit)(t _(m) ,U _(n))=c _(m) ⁽¹⁾ U _(n) +c _(m) ⁽²⁾ U _(n) ² +c _(m)⁽³⁾ U _(n) ³,  (Eq. 3)

where the coefficients c_(m) ^((i)) are obtained by minimizing:

$\begin{matrix}{\xi_{m} = {\sum\limits_{n}^{N}{\left( {{y\left( {t_{m},U_{n}} \right)} - {y_{fit}\left( {t_{m},U_{n}} \right)}} \right)^{2}.}}} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$

Eq. 3 is written as a matrix:

$\begin{matrix}{{{\begin{pmatrix}U_{1} & U_{1}^{2} & U_{1}^{3} \\U_{2} & U_{2}^{2} & U_{2}^{3} \\\vdots & \vdots & \vdots \\U_{N} & U_{N}^{2} & U_{N}^{3}\end{pmatrix}\begin{pmatrix}c_{m}^{(1)} \\c_{m}^{(2)} \\c_{m}^{(3)}\end{pmatrix}} = \begin{pmatrix}{y\left( {,U_{1}} \right)} \\{y\left( {,U_{2}} \right)} \\\vdots \\{y\left( {,U_{N}} \right)}\end{pmatrix}},} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$

or Uc=y. The polynomial coefficient vector c is obtained by a fitting ina least square sense:

c=(U ^(H) U)⁻¹ U ^(H) y,  (Eq. 6)

where U^(H) is the hermitian transpose of the matrix U. Here U is realand U^(H)=U^(T), where U^(T) is the transpose.

Using the linear term, the step response data y(t_(m),U_(n)) isnormalized:

$\begin{matrix}{{y_{norm}\left( {t_{m},U_{n}} \right)} = {\frac{y\left( {t_{m},U_{n}} \right)}{c_{m}^{1}}.}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$

The nonlinear part, i.e., the difference between y_(norm)(t_(m),U_(n))and a straight line, is calculated at each (t_(m),U_(n)):

y _(NL)(t _(m) ,U _(n))=y _(norm)(t _(m) ,U _(n))−U _(n).  (Eq. 8)

The average integrated nonlinearity measure, Avg_INL, is calculated asfollows:

$\begin{matrix}{{Avg\_ INL} = {\frac{1}{MN}{\sum\limits_{m = 0}^{M - 1}{\sum\limits_{n = 1}^{N}{{y_{NL}\left( {t_{m},n} \right)}}}}}} & \left( {{Eq}.\mspace{14mu} 9} \right)\end{matrix}$

Instead of using the non-linear measure INL (or Avg_INL) as discussedabove, a different way to express non-linear properties of the seismicsensing module 200 is by use of a total harmonic distortion (THD)measure in the case that the input test signal is a sinusoidal withangular frequency ω. The output response of the sensing module 200 isconsidered to have signal components at various frequencies, includingthe following angular frequencies: ω, 2ω, 3ω, 4ω, . . . . The signalcomponent at 2ω represents the second order harmonic, the signalcomponent at 3ω represents the third order harmonic, and so forth. TheTHD measure is calculated as the sum of the power at frequencies 2ω, 3ω,4ω, and so forth.

The computation of THD according to one example implementation isdescribed below. The THD, which is a standard measure, could becalculated from the coefficients c₁, c₂, . . . , as determined from thestep test (see Eq. 6). The input test signal can be written as:

x(t)=A cos(ωt),  (Eq. 10)

where A is the amplitude and ω is the angular frequency.

If it is assumed that the non-linear system can be described by a 4thorder polynomial, with the polynomial coefficients c₁, . . . , c₄, thefollowing output signal is obtained:

$\begin{matrix}\begin{matrix}{{y(t)} = {{c_{1}A\; {\cos \left( {\omega \; t} \right)}} + {c_{2}A^{2}{\cos \left( {\omega \; t} \right)}^{2}} + {c_{3}A^{3}{\cos \left( {\omega \; t} \right)}^{3}} +}} \\{{c_{4}A^{4}{\cos \left( {\omega \; t} \right)}^{4}}} \\{= {{c_{1}A\; {\cos \left( {\omega \; t} \right)}} + {\frac{c_{2}A^{2}}{2}\left( {1 + {\cos \left( {2\; \omega} \right)}} \right)} +}} \\{{{\frac{c_{3}A^{3}}{4}\left( {{\cos \left( {3\; \omega \; t} \right)} + {3\; \cos \left( {\omega \; t} \right)}} \right)} +}} \\{{\frac{c_{4}A^{4}}{4}\left( {\frac{3}{2} + {2\; {\cos \left( {2\; \omega \; t} \right)}} + {\frac{1}{2}{\cos \left( {4\; \omega \; t} \right)}}} \right)}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 11} \right)\end{matrix}$

The relative amplitude (in linear scale) at 2ω (neglecting thecontributions from c₆, c₈, . . . ) is as follows:

$\begin{matrix}{{{HD}(w)} = {\frac{c_{2}A}{2c_{1}} + \frac{c_{4}A^{3}}{4c_{1}}}} & \left( {{Eq}.\mspace{14mu} 12} \right)\end{matrix}$

The relative amplitude at 3ω (neglecting the contributions from c₅, c₇,. . . ) is as follows:

$\begin{matrix}{{{HD}\left( {3\omega} \right)} = {\frac{c_{3}A^{2}}{4c_{1}}.}} & \left( {{Eq}.\mspace{14mu} 13} \right)\end{matrix}$

The relative amplitude at 4ω (neglecting the contributions from c₆, c₈,. . . ) is as follows:

$\begin{matrix}{{{HD}\left( {4\; \omega} \right)} = {\frac{c_{4}A^{3}}{8c_{1}}.}} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$

Assuming just the harmonics 2ω, 3ω, and 4ω are considered, the totalharmonic distortion, THD, can then be calulated as:

THD=HD(2ω)+HD(3ω)+HD(4ω),  (Eq. 15)

which is normally given in dB.

Thus, if the polynomial coefficients c₁, c₂, c₃, c₄ are known, the THDcan be calculated for a given amplitude A.

More generally, the non-linear properties of a seismic sensing elementare determined from a number of voltage steps of different amplitudes.The voltage steps are applied to the seismic sensing element, with thestep response recorded in the time domain. The non-linear properties aretaken as the step response's deviation from a straight line as afunction of applied step voltage.

By determining the non-linear properties of the seismic sensing element,distortion in the recorded seismic signal can be removed or reduced.Also, in some cases, if it is detected that a seismic sensing element isexhibiting excessive non-linear properties, then the correspondingseismic sensing module 200 can be removed or disabled from operation orservice.

Determining the non-linear characteristic of the sensing module 200 canbe performed in the field (such as during a land seismic surveyingoperation, a marine seismic surveying operation, or a wellbore seismicsurveying operation). The processor local test circuitry in each seismicsensing module 200 can also be used for production testing of thesensing module 200, which can be cheaper and faster than testing thatinvolves use of an external signal generator.

Techniques according to some embodiments can be applied to differenttypes of seismic sensing elements, including geophone accelerometers,moving coil geophones, velocimeters, microelectro-mechanical systems(MEMS) accelerometers, and so forth.

Polarity Testing

When assembling seismic sensing elements onto one or plural streamers,or in some other seismic spread, which involves the connection ofelectrical wires to the seismic sensing elements, the polarities of wireconnections to the seismic sensing elements can be inadvertentlyreversed. Note that a seismic surveying system can include a relativelylarge number of seismic sensing elements (e.g., thousands, tens ofthousands, hundreds of thousands), such that it would be easy to reversethe wire connections to some of the seismic sensing elements. Reversingthe wire connections to seismic sensing elements would result ininaccurate measurement data being collected from the correspondingseismic sensing elements, which can adversely affect accuracy of theseismic surveying operation.

To avoid having to use an external impulse to perform polarity testing,the self-contained seismic sensing module 200 (FIG. 2) according to someembodiments can be used to perform self-testing to detect for reversedpolarity of the seismic sensing element 204 contained in the sensingmodule 200. In accordance with some embodiments, the processor 210 inthe sensing module 200 is used for applying a positive input testvoltage step and a negative input test voltage step to the seismicsensing element 204. The positive and negative input voltage steps causethe seismic sensing element 204 to produce output signals of differentamplitudes. The amplitudes of the output signals responsive to thepositive and negative input voltage steps depend upon the connectedpolarity of the seismic sensing element 204. Based on these detectedamplitudes, it can be determined whether the polarity of the wireconnections to the seismic sensing element is reversed. By using theinternal processor 210 to perform the polarity testing, no externalimpulse source is required.

FIG. 6A shows the seismic sensing element 204 having a first polarity(referred to as a normal polarity), with the positive terminal 602 ofthe seismic sensing element 204 connected to a first wire 604, and anegative terminal 606 of the seismic sensing element 204 connected to asecond wire 608. The input voltage U is defined from the first wire 604to the second wire 608.

FIG. 6B shows the polarity of the seismic sensing element 204 reversed,with the negative terminal 606 connected to the first wire 604, and thepositive terminal 602 connected to the second wire 608. It is desired toautomatically identify, using the processor 210 of the self-containedseismic sensing module 200, the reversed polarity connection depicted inFIG. 6B.

FIG. 7A shows a curve 702 that represents the step response, y(t),responsive to a positive step input voltage (which is +0.7 volts in theexample of FIG. 7A). FIG. 7A also shows a second curve 704 thatrepresents the step response, y(t), that is responsive to a negativestep input voltage (e.g., −0.7 volts). FIG. 7A shows the expectedresponses to the positive and negative input steps when the polarity ofthe seismic sensing element 204 is correct (the polarity depicted inFIG. 6A).

FIG. 7B shows two curves 706, 708, with curve 706 representing the stepresponse, y(t), that is responsive to an applied negative input step,and the curve 708 representing the response, y(t), that is responsive toan applied positive input step. FIG. 7B shows the positive and negativestep responses for the seismic sensing element 204 whose polarity hasbeen reversed (FIG. 6B). Thus, as can be seen from a comparison of FIGS.7A and 7B, the negative step response (706) when the polarity isreversed is much greater than the negative step response when thepolarity is normal. Similarly, the positive step response (708) when thepolarity is reversed is much smaller than the negative step response(702) when the polarity is normal.

The curves of FIG. 7A can be considered expected profiles of thepositive and negative step responses, where the expected profiles can bestored in the storage 218 of the sensing module 200. If the processor210 determines that the positive and negative step responses of a givenseismic sensing element generally match the expected profiles, then theprocessor 210 produces an indication that the polarity of the seismicsensing element 204 is correct. On the other hand, if the processor 210detects a substantial difference (difference by greater thanpredetermined one or more thresholds) between the positive and negativestep responses of the seismic sensing element 204 and the expectedprofiles, then the processor 210 can produce a second indication toindicate that the polarity of the seismic sensing element 204 has beenreversed. The indication can be a Boolean indication, where a firststate indicates correct polarity, and a second state indicates reversedpolarity.

Alternatively, instead of comparing a measured step response to expectedprofiles, a simple summation of the positive and negative step responsecan be performed, such as according to Eq. 16 below, to produce anindication of correct polarity or reversed polarity.

The positive step response can be represented as y⁺(t), where trepresents time, while the negative step response can be represented asy⁻(t). A number p can be calculated as follows:

$\begin{matrix}{{p = {\sum\limits_{t}^{\;}\left( {{y^{+}(t)} - {y^{-}(t)}} \right)}},} & \left( {{Eq}.\mspace{14mu} 16} \right)\end{matrix}$

where p is a sum of the difference between the positive step responseand the negative step response at plural time points t. If p is greaterthan zero (p>0) or some other threshold, then the polarity is correct.However, if p is less than zero or some other threshold, (p<0), then thepolarity is reversed.

It is noted that in some embodiments, the magnitude of the input stepvoltage (both the magnitude of the positive input and the magnitude ofthe negative input step) should be greater than some predeterminedthreshold. In the example of FIGS. 7A and 7B, the magnitude of thepositive and negative input steps is 0.7 volts. In some cases, if themagnitude of the positive and negative input steps drops below thepredetermined threshold, such as if the magnitude is 0.3 volts, then thepositive and negative step responses may not differ by much for thedifferent polarities of the seismic sensing element 204, which can makedetection of reversed polarity difficult. This is illustrated in theexample of FIGS. 8A and 8B, where the curves indicate small variationsbetween the positive and negative step responses for differentpolarities of the seismic sensing element 204.

Determining Transfer Function of the Seismic Sensing Module

The step response of the seismic sensing module 200 in response to aninput test step signal is defined according to a transfer function ofthe seismic sensing module 200. The transfer function, in the frequencydomain, has a first order approximation as follows:

$\begin{matrix}{{{H(\omega)} = \frac{Y(\omega)}{A^{ref}(\omega)}},} & \left( {{Eq}.\mspace{14mu} 17} \right)\end{matrix}$

where A^(ref)(ω) corresponds to the signal that would produce the outputsignal Y(ω) (frequency domain representation of y(t)) for a given H(ω)).The transfer function can be determined from the measured step responseusing either a non-parametric method or a parametric method. Aparametric method is needed if the geophones have geophone parametersthat vary significantly between different units. The non-parametricmethod is more suitable if different geophones have transfer functionsthat do not differ significantly, except for when they have differenttemperature.

For enhanced accuracy, additional terms are added to the transferfunction definition according to Eq. 17, where the additional terms areused to compensate for changes of the transfer function due to factorssuch as changes in temperature and/or other factors.

Y(ω) is the Fourier transform (FFT) of the measured step response,Y(ω)=FFT[y(t)]. A^(ref)(ω) is a reference acceleration, which includestabulated values obtained from experiments or possibly simulations onthe sensing module. Notice that A^(ref)(ω) has to be determined fromknown Y(ω) and H(ω), and that A^(ref)(ω) is not a step in acceleration.Rather, A^(ref)(ω) is stimuli that would cause the output signal Y(ω),if the system were time invariant.

FIGS. 9A and 9B depict A^(ref)(ω) that is determined fromexperimentation on the sensing module. FIG. 9A shows the magnitude ofA^(ref) as a function of frequency (curve 802), and FIG. 9B shows thephase of A^(ref) as a function of frequency (curve 804). The magnitudecurve 802 and phase curve 804 are contrasted to the ideal dotted curves806 and 808 in FIGS. 9A and 9B, respectively, which depict the Fouriertransform of an ideal step in acceleration (assuming that the seismicsensing element is an accelerometer). Note that the technique ofidentifying A^(ref) can also be applied to other types of sensingelements, such as velocimeters and MEMS accelerometers.

As noted above, the transfer function H(ω) is dependent upon externalfactors such as temperature, such that Eq. 17 may be inaccurate in somescenarios. To add terms to Eq. 17, the following definition of H(ω) isprovided:

$\begin{matrix}{{{H(\omega)} = {\frac{Y(\omega)}{A^{ref}(\omega)} + {F_{r}\left\lbrack {Y(\omega)} \right\rbrack} + {j\; {F_{j}\left\lbrack {Y(\omega)} \right\rbrack}}}},} & \left( {{Eq}.\mspace{14mu} 18} \right)\end{matrix}$

where F_(r)[Y(ω)] and jF_(j)[Y(ω)] are temperature-dependent terms,

F _(r) [Y(ω)]=c _(r)(ω)(Y _(r)(ω)−Y _(r) ^(ref)(ω))+c _(j)(ω)(Y _(i)^(ref)(ω))+c _(rr)(ω)(Y _(r)(ω)−Y _(r) ^(ref)(ω))² +c _(jj)(ω)(Y_(i)(ω)−Y _(i) ^(ref)(ω))² +c _(rj)(ω)Y _(r)(ω)−Y _(r) ^(ref)(ω))(Y_(i)(ω)−Y _(i) ^(ref)(ω))+c _(rrr)(ω)(Y _(r)(ω)−Y _(r) ^(ref)(ω))³ +c_(jjj)(ω)(Y _(i)(ω)−Y _(r) ^(ref)(ω))³ +c _(rrj)(ω)(Y _(r)(ω)−Y _(r)^(ref)(ω))²(Y _(i)(ω)−Y _(i) ^(ref)(ω))+c _(rrj)(ω)(Y _(r)(ω)−Y _(r)^(ref)(ω))(Y _(i)(ω)−Y _(i) ^(ref)(ω))² +D(ω),

and

F _(r) [Y(ω)]=d _(r)(ω)(Y _(r)(ω)−Y _(r) ^(ref)(ω))+d _(j)(ω)(Y _(i)^(ref)(ω))+d _(rr)(ω)(Y _(r)(ω)−Y _(r) ^(ref)(ω))² +d _(jj)(ω)(Y_(i)(ω)−Y _(i) ^(ref)(ω))² +d _(rj)(ω)Y _(r)(ω)−Y _(r) ^(ref)(ω))(Y_(i)(ω)−Y _(i) ^(ref)(ω))+d _(rrr)(ω)(Y _(r)(ω)−Y _(r) ^(ref)(ω))³ +d_(jjj)(ω)(Y _(i)(ω)−Y _(r) ^(ref)(ω))³ +d _(rrj)(ω)(Y _(r)(ω)−Y _(r)^(ref)(ω))²(Y _(i)(ω)−Y _(i) ^(ref)(ω))+d _(rrj)(ω)(Y _(r)(ω)−Y _(r)^(ref)(ω))(Y _(i)(ω)−Y _(i) ^(ref)(ω))² +D(ω),

Y_(i)(ω) and Y_(r)(ω) are the real and imaginary parts of the measuredstep response during a step test, Y(ω). Y_(r) ^(ref)(ω) and Y_(i)^(ref)(ω) are the real and imaginary parts of a reference step response.c_(r)(ω), c_(j)(ω), . . . , d_(r)(ω), d_(j)(ω) are constants that areobtained from simulations or measurements of the temperature dependenceof the sensing element's transfer function.

The equations above describe a Taylor series in the real and imaginaryparts around an operating point. Notice that Y_(r) ^(ref)(ω) and Y_(i)^(ref)(ω) are determined from experiments and are then tabulated. Thecoefficients c_(r)(ω), c_(j)(ω) , . . . , d_(r)(ω), d_(j)(ω) are alsodetermined from experiments or simulations and are tabulated. The numberof tabulated values will be relatively high. However, since thefrequency dependence is relatively weak, Y_(r) ^(ref)(ω) and Y_(i)^(ref) (ω), and c_(r)(ω) . . . can be tabulated at a few frequencies andthe data can then obtained by interpolation or compressed in other wayssuch as polynomials or other basis function that are functions offrequency.

Thus, A^(ref)(ω), F_(r)[Y(ω)], F_(j)[Y(ω)] can be determined fromsimulations or experiments. These functions are stored as tabulatedvalues in the storage 218 of the sensing module 200. During a testingprocedure, Eq. 18 can be used to determine H(ω) once the step responsein the frequency domain, Y(ω), is obtained. The transfer function canthen be used for performing seismic data correction, also referred to asequalization. Seismic data correction can be performed in two steps:first the transfer function is corrected (for differences compared to areference one); secondly, the seismic data are corrected using thecorrected transfer function.

As explained further below, the temperature-dependent transfer function(H(ω)), derived from the step test, can be used to produce an inversefilter H⁻¹(ω) that can be used for performing correction on acquiredseismic data during a seismic operation. Note that thetemperature-dependent H(ω) can be updated before each seismic operationto improve accuracy in seismic data collection.

FIGS. 10A and 10B depict the real and imaginary parts of

${H(\omega)} - \frac{Y(\omega)}{A(\omega)}$

at a specific frequency, ω=30 hertz. The vertical axis in FIG. 10A showsthe real part of

${{H(\omega)} - \frac{Y(\omega)}{A(\omega)}},$

while the vertical axis of FIG. 10B shows the imaginary part of

${H(\omega)} - {\frac{Y(\omega)}{A(\omega)}.}$

The two horizontal axes in FIGS. 10A and 10B are the real part andimaginary part of Y(ω).

Each curve 902 and 904 in respective FIG. 10A and FIG. 10B correspondsto changes in temperature from a first temperature to a secondtemperature. Note that the real and imaginary parts of

${H(\omega)} - \frac{Y(\omega)}{A(\omega)}$

correspond to F_(r) and F_(j) in Eq. 18 above.

Another way of determining the transfer function is to use a parametrictechnique, which is described below. The described method below isdescribed for the geophone accelerometer. It could be used also for atraditional step test of a geophone.

The method is based on a linearization of the analytical expressions forthe step response around nominal values of the geophone parameters. Itworks if the geophone parameters are close to the nominal values. Thereason for using this method is that the geophone parameters can beestimated (or determined) using linear methods which can be implementedin digital signal processor (or FPGAs). If the expressions are notlinearized, large computer memory and time may be required.

Assume that the transfer function is given as function of the angularfrequency ω, and the geophone parameters S (sensitivity), ω₀ (naturalfrequency), D (damping), Z (DC resistance) and m (mass). For the nominalor reference values of the geophone parameters, a reference transferfunction H_(ref)(ω) is defined as follows:

H _(ref)(ω)=H(ω,S,ω ₀ ,D,Z,m).  (Eq. 19)

Taylor's formula gives that:

$\begin{matrix}{{{H\left( {\omega,{S + {\Delta \; S}},{\omega_{0} + {\Delta \; \omega_{0}}},{D + {\Delta \; D}},{Z + {\Delta \; Z}},{m + {\Delta \; m}}} \right)} = {{H_{ref}(\omega)} + {\Delta \; S\frac{\partial\;}{\partial S}{H_{ref}(\omega)}} + {\Delta \; \omega_{0}\frac{\partial\;}{\partial\omega_{0}}{H_{ref}(\omega)}} + {\Delta \; D\frac{\partial\;}{\partial D}{H_{ref}(\omega)}} + {\Delta \; S\frac{\partial\;}{\partial S}{H_{ref}(\omega)}} + {\Delta \; m\frac{\partial\;}{\partial m}{H_{ref}(\omega)}}}},} & \left( {{Eq}.\mspace{14mu} 20} \right)\end{matrix}$

where it is assumed that ΔS/S<<1, etc.

In a similar way, the step response is a function of the sameparameters:

Y _(ref)(ω)=Y(ω,S,ω ₀ ,D,Z,m)  (Eq. 21)

Taylor's formula gives:

$\begin{matrix}{{Y\left( {\omega,{S + {\Delta \; S}},{\omega_{0} + {\Delta \; \omega_{0}}},{D + {\Delta \; D}},{Z + {\Delta \; Z}},{m + {\Delta \; m}}} \right)} = {{Y\left( {\omega,S,\omega_{0},D,Z,m} \right)} + {\Delta \; S\frac{\partial\;}{\partial S}{Y\left( {\omega,S,\omega_{0},D,Z,m} \right)}} + {\Delta \; \omega_{0}\frac{\partial\;}{\partial\omega_{0}}{Y\left( {\omega,S,\omega_{0},D,Z,m} \right)}} + {\Delta \; D\frac{\partial\;}{\partial D}{Y\left( {\omega,S,\omega_{0},D,Z,m} \right)}} + {\Delta \; Z\frac{\partial\;}{\partial Z}{Y\left( {\omega,S,\omega_{0},D,Z,m} \right)}} + {\Delta \; m\frac{\partial\;}{\partial m}{Y\left( {\omega,S,\omega_{0},D,Z,m} \right)}}}} & \left( {{Eq}.\mspace{14mu} 22} \right)\end{matrix}$

Y_(exp)(ω) is determined experimentally, and is defined as:

Y _(exp)(ω)=Y(ω,S+ΔS,ω ₀+Δω₀ ,D+ΔD,Z+ΔZ,m+Δm)  (Eq. 23)

The difference can be written as:

$\begin{matrix}\begin{matrix}{{\Delta \; {Y(\omega)}} = {{Y_{\exp}(\omega)} - {Y_{ref}(\omega)}}} \\{= {{\Delta \; S\frac{\partial\;}{\partial S}{Y_{ref}(\omega)}} + {\Delta \; \omega \frac{\partial\;}{\partial\omega}{Y_{{ref}\;}(\omega)}} +}} \\{{{\Delta \; D\frac{\partial\;}{\partial F}{Y_{ref}(\omega)}} + {\Delta \; Z\frac{\partial\;}{\partial Z}{Y_{ref}(\omega)}} +}} \\{{\Delta \; m\frac{\partial\;}{\partial m}{Y_{ref}(\omega)}}}\end{matrix} & \left( {{Eq}.\mspace{14mu} 24} \right)\end{matrix}$

The real and imaginary part can be written, respectively, as:

$\begin{matrix}{{{{Re}\left\lbrack {\Delta \; {Y(\omega)}} \right\rbrack} = {{\Delta \; S\; {{Re}\left\lbrack {\frac{\partial\;}{\partial S}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; \omega \; {{Re}\left\lbrack {\frac{\partial\;}{\partial\omega}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; D\; {{Re}\left\lbrack {\frac{\partial\;}{\partial D}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; Z\; {{Re}\left\lbrack {\frac{\partial\;}{\partial Z}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; m\; {{Re}\left\lbrack {\frac{\partial\;}{\partial m}{Y_{ref}(\omega)}} \right\rbrack}}}},} & \left( {{Eq}.\mspace{14mu} 25} \right) \\{{{Im}\left\lbrack {\Delta \; {Y(\omega)}} \right\rbrack} = {{\Delta \; S\; {{Im}\left\lbrack {\frac{\partial\;}{\partial S}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; \omega \; {{Im}\left\lbrack {\frac{\partial\;}{\partial\omega}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; D\; {{Im}\left\lbrack {\frac{\partial\;}{\partial D}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; Z\; {{Im}\left\lbrack {\frac{\partial\;}{\partial Z}{Y_{ref}(\omega)}} \right\rbrack}} + {\Delta \; m\; {{{Im}\left\lbrack {\frac{\partial\;}{\partial m}{Y_{ref}(\omega)}} \right\rbrack}.}}}} & \left( {{Eq}.\mspace{14mu} 26} \right)\end{matrix}$

Data at frequencies ω₁, ω₂, . . . , ω_(M), can be used, and m≧5 (sincethere are five parameters). The above can be written as an equationsystem in matrix form:

$\begin{matrix}{\begin{bmatrix}{{Re}\left\lbrack {\Delta \; {Y\left( \omega_{1} \right)}} \right\rbrack} \\\vdots \\{{Re}\left\lbrack {\Delta \; {Y\left( \omega_{M} \right)}} \right\rbrack} \\{{Im}\left\lbrack {\Delta \; {Y\left( \omega_{1} \right)}} \right\rbrack} \\\vdots \\{{Im}\left\lbrack {\Delta \; {Y\left( \omega_{M} \right)}} \right\rbrack}\end{bmatrix}{\quad{\quad{{\left\lbrack \begin{matrix}{{Re}\left\lbrack {\frac{\partial\;}{\partial S}{Y_{ref}\left( \omega_{1} \right)}} \right\rbrack} & {{Re}\left\lbrack {\frac{\partial\;}{\partial\omega}{Y_{ref}\left( \omega_{1} \right)}} \right\rbrack} & \ldots & {{Re}\left\lbrack {\frac{\partial\;}{\partial m}{Y_{ref}\left( \omega_{1} \right)}} \right\rbrack} \\\vdots & \; & \; & \; \\{{Re}\left\lbrack {\frac{\partial\;}{\partial S}{Y_{ref}\left( \omega_{M} \right)}} \right\rbrack} & \; & \; & \vdots \\{{Im}\left\lbrack {\frac{\partial\;}{\partial S}{Y_{ref}\left( \omega_{1} \right)}} \right\rbrack} & \; & \; & \vdots \\{{Im}\left\lbrack {\frac{\partial\;}{\partial S}{Y_{ref}\left( \omega_{M} \right)}} \right\rbrack} & {{Im}\left\lbrack {\frac{\partial\;}{\partial\omega}{Y_{ref}\left( \omega_{M} \right)}} \right\rbrack} & \ldots & {{Im}\left\lbrack {\frac{\partial\;}{\partial m}{Y_{ref}\left( \omega_{M} \right)}} \right\rbrack}\end{matrix} \right\rbrack \begin{bmatrix}{\Delta \; S} \\{\Delta \; \omega_{0}} \\\vdots \\{\Delta \; m}\end{bmatrix}},}}}} & \left( {{Eq}.\mspace{14mu} 27} \right)\end{matrix}$

or simply,

E=DP.  (Eq. 28)

The parameters given in the vector P can now be determined in the leastsquare sense:

P=(D ^(H) D)⁻¹ D ^(H) E,  (Eq. 29)

where H denotes the hermitian transpose.

The determined parameters P=[ΔS Δω₀ . . . Δm]^(T) can be used in Eq. 20to estimate the transfer function.

The expressions for

${\frac{\partial\;}{\partial S}{H_{ref}(\omega)}},{\frac{\partial\;}{\partial\omega_{0}}{H_{ref}(\omega)}},{\frac{\partial\;}{\partial D}{H_{ref}(\omega)}},{\frac{\partial\;}{\partial S}{H_{ref}(\omega)}},{\frac{\partial\;}{\partial m}{H_{ref}(\omega)}}$and${\frac{\partial\;}{\partial S}{Y(\omega)}},{\frac{\partial\;}{\partial\omega_{0}}{Y(\omega)}},{\frac{\partial\;}{\partial D}{Y(\omega)}},{\frac{\partial\;}{\partial Z}{Y(\omega)}},{\frac{\partial\;}{\partial m}{Y(\omega)}}$

are determined and stored in memory. These could be determined from ananalytical expression of H_(ref)(ω) and Y_(ref)(ω) or could be estimatedfrom circuit simulations. For example,

${{\frac{\partial\;}{\partial S}{H_{ref}(\omega)}} \approx {\frac{\,^{\prime}1}{\Delta \; S_{ref}}\left( {{H\left( {\omega,{S + {\Delta \; S}},\omega_{0},D,Z,m} \right)} - {H\left( {\omega,S,\omega_{0},D,Z,m} \right)}} \right)}},$

where ΔS_(ref) is a small change in around the nominal value S.H(ω,S,+ΔS,ω₀,D,Z,m) and H(ω,S,ω₀,D,Z,m), can be calculated from thesimulations.

Auto-Calibration

In some implementations, sensitivity correction can be performed by theprocessor 210 (FIG. 1) in a self-contained sensing module. To a firstapproximation, the sensing module response can be corrected by applyinga fixed correction factor to match a reference response, represented asH_(ref)(ω), where H_(ref)(ω) represents the reference transfer functionthat defines the reference response. By performing such correction, thesensitivity of the sensing module over a defined frequency bandwidth canbe improved.

The reference model, represented as H_(ref)(ω), can be stored intabulated form in a table in the storage 218 of FIG. 2. The correctionfactor can be computed by a least mean square technique, according to anembodiment.

The frequency dependence sensitivity correction factor, S_(err)(ω), canbe calculated as follows:

$\begin{matrix}{{\frac{{H_{ref}(\omega)}}{{H_{est}(\omega)}} = {1 + {S_{err}(\omega)}}},} & \left( {{Eq}.\mspace{14mu} 30} \right)\end{matrix}$

where H_(est)(ω) is the transfer function determined according to Eq. 18discussed above.

From Eq. 30, the following can be obtained:

$\begin{matrix}{{S_{err}(\omega)} = {\left( {\frac{{H_{ref}(\omega)}}{{H_{est}(\varpi)}} - 1} \right).}} & \left( {{Eq}.\mspace{14mu} 31} \right)\end{matrix}$

From different values of S_(err)(ω) at different frequencies, an averagesensitivity correction factor, S_(cal), is obtained as follows:

$\begin{matrix}{{{S_{cal} = {\sum\limits_{i = 1}^{L}{{v\left( \omega_{i} \right)}{{S_{err}\left( \omega_{i} \right)}/\left( {\sum\limits_{i = 1}^{L}{v\left( \omega_{i} \right)}} \right)}}}},{where}}{{v\left( \omega_{i} \right)} = {\frac{1}{\omega_{i}}.}}} & \left( {{Eq}.\mspace{14mu} 32} \right)\end{matrix}$

In other implementations, other spectra weighting functions can be used.The correction factor, S_(cal), can then be applied to a continuous flowof data in real time to allow a better match from sensing module tosensing module. This technique corrects for the amplitude of thetransfer function, but not for its frequency dependence.

In another implementation, instead of correcting for just the amplitudeof the transfer function, a complete transfer function correction can beperformed as follows. To do so, a coefficient of a matching filter canbe computed to adapt the transfer function to a reference model over thecomplete bandwidth of interest. Using this technique, the inverse of thetransfer function is first determined. This technique is commonlyreferred to as equalization. If the frequency domain input to thesensing module is X(ω), the transfer function is H(ω), and the outputsignal is Y(ω), then the inverse filter (equalizer) is denoted H⁻¹(ω).The signal after the inverse filter, H⁻¹(ω) is denoted Y′(ω).Consequently, Y′(ω) is computed as follows:

Y(ω)=H(ω)X(ω),

Y′(ω)=H ⁻¹(ω)H(ω)X(ω)=X(ω).  (Eq. 33)

Notice that H⁻¹(ω)=1/H(ω). In this case, H(ω) is determined from thestep test (Eq. 18), and thus H⁻¹(ω) can be readily determined. Applyingthe inverse filter H⁻¹(ω) to the acquired seismic data collected by thecorresponding seismic sensing element provides transfer functioncorrection on the seismic data. Note that the inverse filter H⁻¹(ω) istemperature dependent. If the temperature-dependent transfer functionH(ω) is updated before each seismic operation (e.g., once in a morningoperation and once in an afternoon operation), then H(ω), and thusH⁻¹(ω), will reflect temperature fluctuations in the system to enablethe seismic sensing module to produce more accurate data.

Standard ways of determining a casual digital filter (in the timedomain) from a known frequency response can be used. The filter can beeither IIR or FIR. Standard methods like the bilinear transformation canalso be used.

Another calibration that can be performed by the sensing module iscorrection for non-linear distortion (where determination of non-linearproperties of a seismic sensing module is discussed above). To do so,the inverse of the non-linear transfer function is first determined.Using the Volterra theory, the non-linear transfer function (in timedomain) can be written as:

y(t)=H[x(t)]=H ₁ [x(t)]+H ₂ [x(t)]+H ₃ [x(t)],

and the inverse

y′(t)=H ⁻¹ [y(t)]=H ⁻¹ [H[x(t)]]=x(t)

y′(t)=H ⁻¹ [y(t)]=H ₁ ⁻¹ [y(t)]+H ₂ ⁻¹ [y(t)]+H ₃ ⁻¹ [y(t)]+ . . .

Since the seismic system is a weakly linear system with small frequencydependence, H⁻¹ can be readily performed. An advantage of correcting fornon-linear distortion is that the signal becomes less distorted. Anotheradvantage is that the dynamic amplitude range of the acquisition changemay increase.

The various tasks discussed above can be performed by a local processorin each seismic sensing module. In other implementations, at least oneof the tasks can be performed by software which can be loaded forexecution on a processor. The processor includes microprocessors,microcontrollers, processor modules or subsystems (including one or moremicroprocessors or microcontrollers), or other control or computingdevices. A “processor” can refer to a single component or to pluralcomponents.

Data and instructions (of the software) are stored in respective storagedevices, which are implemented as one or more computer-readable orcomputer-usable storage media. The storage media include different formsof memory including semiconductor memory devices such as dynamic orstatic random access memories (DRAMs or SRAMs), erasable andprogrammable read-only memories (EPROMs), electrically erasable andprogrammable read-only memories (EEPROMs) and flash memories; magneticdisks such as fixed, floppy and removable disks; other magnetic mediaincluding tape; and optical media such as compact disks (CDs) or digitalvideo disks (DVDs).

While the invention has been disclosed with respect to a limited numberof embodiments, those skilled in the art, having the benefit of thisdisclosure, will appreciate numerous modifications and variationstherefrom. It is intended that the appended claims cover suchmodifications and variations as fall within the true spirit and scope ofthe invention.

1. A method of testing a seismic sensing module, comprising: generating,using a local processor associated with a seismic sensing element in theseismic sensing module, a test signal that is applied to the seismicsensing element; receiving, by the local processor, a response of theseismic sensing element to the test signal; and determining, by thelocal processor, at least one characteristic of the seismic sensingmodule selected from a polarity of the seismic sensing element in theseismic sensing module, a non-linear property of the seismic sensingmodule, and a temperature-dependent transfer function of the seismicsensing module.
 2. The method of claim 1, wherein determining thepolarity of the seismic sensing element in the seismic sensing modulecomprises: comparing the response of the seismic sensing element to thetest signal to an expected profile, the method further comprising:indicating a correct polarity in response to the response substantiallymatching the expected profile; and indicating a reversed polarity inresponse to the response not matching the expected profile.
 3. Themethod of claim 1, wherein determining the polarity of the seismicsensing element in the seismic sensing module comprises: receiving afirst response of the seismic sensing element in response to the testsignal that has a positive polarity; and receiving a second response ofthe seismic sensing element in response to another test signal that hasa negative polarity.
 4. The method of claim 3, further comprisingcombining the first response and the second response to produce anindication, wherein the indication has a first value to indicate acorrect polarity of the seismic sensing element, and a second value toindicate an incorrect polarity of the seismic sensing element.
 5. Themethod of claim 4, wherein the first value is a value less than apredetermined threshold, and wherein the second value is a value greaterthan the predetermined threshold.
 6. The method of claim 1, whereindetermining the non-linear property of the seismic sensing modulecomprises: comparing measured data points corresponding to the responseto a linear response of the seismic sensing module.
 7. The method ofclaim 6, wherein comparing the measured data points to the linearresponse comprises computing a sum of differences between the measureddata points and the linear response for different time points anddifferent step voltages of the test signal.
 8. The method of claim 7,further comprising dividing the sum by a number of the step voltages andby a number of the time points to produce a measure of the non-linearityof the seismic sensing module.
 9. The method of claim 1, whereindetermining the non-linear property of the seismic sensing modulecomprises calculating a total harmonic distortion.
 10. The method ofclaim 1, wherein determining the temperature-dependent transfer functionof the seismic sensing module comprises providing terms in the transferfunction that are temperature dependent.
 11. The method of claim 10,further comprising determining the terms based on simulations orexperiments.
 12. The method of claim 10, wherein thetemperature-dependent transfer function is one of a parametric transferfunction and a non-parametric transfer function.
 13. The method of claim1, further comprising producing a transfer function that accounts forthe non-linear property of the seismic sensing module, the methodfurther comprising producing an inverse filter based on the transferfunction for applying on seismic data collected by the seismic sensingmodule during a seismic survey operation to correct for non-linearity ofthe seismic sensing module.
 14. The method of claim 1, furthercomprising communicating an indication of an incorrect polarity of theseismic sensing element to a control system.
 15. The method of claim 1,further comprising disabling the seismic sensing element in response todetecting that the seismic sensing element has an incorrect polarity.16. A self-contained seismic sensing module having a single housing thatcontains: a seismic sensing element; a processor; and an interfacecoupled to the processor and to be coupled to an external network,wherein the processor is configured to: generate a test signal appliedto the seismic sensing element, receive a response from the seismicsensing element, and determine a characteristic of the seismic sensingmodule according to the response.
 17. The self-contained seismic sensingmodule of claim 16, wherein the determined characteristic comprises apolarity of wired connections to the seismic sensing element.
 18. Theself-contained seismic sensing module of claim 16, wherein thedetermined characteristic comprises a non-linear property of the seismicsensing module.
 19. The self-contained seismic sensing module of claim16, wherein the determined characteristic comprises atemperature-dependent transfer function of the seismic sensing module.20. The self-contained seismic sensing module of claim 16, wherein thegenerated test signal comprises a step input applied to the seismicsensing element, and wherein the received response comprises a stepresponse.
 21. The self-contained seismic sensing module of claim 16,further comprising a temperature sensor in the housing.
 22. A seismicsensing module comprising: a seismic sensing element; and a processorconfigured to: generate a test signal applied to the seismic sensingelement, receive a response from the seismic sensing element; anddetermine at least one characteristic of the seismic sensing moduleaccording to the response, wherein the at least one characteristic isselected from a polarity of wired connections to the seismic sensingelement, a non-linear property of the seismic sensing module, and atemperature-dependent transfer function of the seismic sensing module.23. The seismic sensing module of claim 22, further comprising a networkinterface for communication over an external network, wherein theprocessor is configured to communicate information relating to the atleast one characteristic over the external network.